I am confused about the absolute value problems. I cannot seem to grasp in my mind how to define this problem piecewised. Any help to get me started would be appreciated.

Actually, there is no problem with the absolute value in this problem! |x| "breaks" when x= 0 so |x+2| "breaks" when x+ 2= 0 or x= -2. Since here you are taking the limit as x goes to 2, not -2, for x "close to 2" (larger than -2) x+2> 0 and this is exactly like
[tex]\lim_{x\rightarrow 2} (x+3)*(x+2)/(x+2)= \lim_{x\rightarrow 2} x+ 3= 2+ 3= 5[/tex].

Now, if the limit were being taken as x goes to -2, THEN you would need to worry about "one sided" limits. No matter how close x is to -2, it could still be either < -2 or > -2.
If x< -2, then x+ 2 is negative and so |x+2|= -(x+2). |x+2|/(x+2)= -(x+2)/(x+2)= -1 so
(x+3)|x+2|/(x+2)= -(x+3) and the limit, as x goes to -2 from below, is -(-2+3)= -1.

But if x> -2, then x+2>0 so |x+2|= x+2 and |x+2|/(x+2)= 1 so (x+3)|x+2|/(x+2)= x+ 3 and the limit, as x goes to -2 from above, is (-2+3)= 1.